pCoMole: Pareto-Constrained Molecule Editing with Discrete Flows

1 Introduction

• Editing known biomolecules must balance multiple properties with biochemical and manufacturability constraints, and shrinkage can support synthesis, manufacturing, and delivery.The pressure is especially acute for large genome editors, where size poses a barrier to viral delivery and in vivo translation.Source: §1 · pp. 1–2
• RayGun and SCISOR [13] address protein shrinkage, but are not designed to jointly optimize multiple objectives under explicit feasibility constraints.RayGun uses fixed-dimensional embeddings and decoding; SCISOR plans deletions through a discrete diffusion process.Source: §1 · p. 2
• Evolutionary search and reinforcement learning can approximate Pareto fronts, while controllable discrete generators generally lack iterative editing with insertions and deletions.The former often rely on repeated online oracle evaluations; controllable generators tend to address constraint satisfaction or preference control separately.Source: §1 · p. 2
• pCoMole steers a pretrained Edit Flow toward feasible, preference-ranked outcomes using an augmented Tchebycheff utility, a Doob-h transform, and short Monte Carlo rollouts.The paper reports improved feasibility and trade-off control versus shrinkage and editing methods, and experimentally confirms retained green fluorescence in two shortened eGFP designs.Source: §1 · p. 2

2 Background

• A continuous-time Markov chain over sequence states is specified by time-inhomogeneous transition rates, whose first-order expansion determines short-time transition probabilities.The time marginals obey the Kolmogorov forward equation under valid rates.Source: §2 · p. 2
• Edit Flows model variable-length sequences with a rate field supported only on single-edit transitions: insertions, deletions, and substitutions.Each edit type is parameterized by a total rate and, where applicable, a token distribution.Source: §2 · p. 3
• Edit Flow training uses an auxiliary alignment with blank symbols to resolve ambiguity among edit paths, yielding a tractable rate-supervision signal.Mismatch coordinates in intermediate alignments imply single-edit updates to the projected sequence.Source: §2 · p. 3

3.1 Problem Setup

• Only terminal sequences must satisfy the constraints; transient intermediate states may leave the feasible set so coordinated edit trajectories remain possible.Infeasible terminal states receive zero preference in the terminal rollouts.Source: §3.1 · p. 3
• The augmented Tchebycheff utility is strictly monotone with respect to Pareto dominance for positive weights, encouraging improvements across objectives rather than a single criterion.Feasibility is incorporated into the terminal preference function, which reweights base Edit Flow samples.Source: §3.2 · p. 4
• With the harmonic function available, the Doob-h guided process samples exactly from the preference-tilted terminal distribution under the stated integrability and positivity assumptions.Changing the weight vector yields different Pareto trade-offs while remaining anchored to the pretrained Edit Flow dynamics.Source: §3.2 · p. 5

3.3 Practical Estimation with Short Rollouts

• Because the harmonic function is intractable in large edit spaces, pCoMole estimates it using short Monte Carlo continuations from candidate edits.Exact simulations of the base CTMC give an unbiased estimator, with feasibility incorporated through the terminal preference function.Source: §3.3 · p. 5
• The practical procedure screens candidate edits, allocates rollouts selectively, retains the best observed feasible terminal, and returns the input if a run finds none.Candidates with only infeasible rollouts receive zero estimated continuation value; the final output is the best feasible terminal observed across the sampling process.Source: §3.3 · p. 5

4.1 Edit Flows Enable High-Quality Biological Sequence Editing

• Five variable-length Edit Flows served as proposal processes, and generated UniRefS sequences closely matched the test sequences’ mean-pLDDT distribution.The trained editors covered short and long proteins, GFP, Cas9, and peptidomimetics; the pLDDT comparison used 1,000 generated and 1,000 test sequences.Source: §4.1 · p. 6

4.2 pCoMole Shrinks GFP While Optimizing Optical Functions Under Experimentally Motivated Constraints

• At 213 residues, pCoMole produced substantially better predicted GFP brightness and excitation alignment than RayGun and SCISOR, including with a UniRef-pretrained editor.The comparison generated 100 candidates per method; all methods had validity near 100%.Source: Table 2 · p. 7, §4.2 · p. 7
• As GFP quality thresholds tightened, pCoMole reached the medium tier faster than SCISOR and alone reached the high tier within the 1-hour budget.SCISOR reached the low tier faster; tiers combine brightness and excitation-error thresholds.Source: Table 3 · p. 7, §4.2 · pp. 7–8

4.5 Ablation Studies

• Joint GFP brightness and excitation guidance gave the best overall trade-off; removing either objective reduced its corresponding optical performance.The ablation used a soft length constraint limiting shrinkage to at most 25 residues; Appendix D also examines constraint, scalarization, deletion-rate, insertion, and sampling settings.Source: §4.2 · p. 8, §4.5 · p. 10

4.2 pCoMole Shrinks GFP While Optimizing Optical Functions Under Experimentally Motivated Constraints

• Two 229-residue eGFP designs with 10 deletions retained clear green fluorescence, unlike a manually selected 228-residue negative control.The pCoMole designs had one or two substitutions and were imaged in BL21 cells at 488 nm.Source: Figure 2 · p. 8, §4.2 · p. 8
• Across four Cas9 orthologs, pCoMole generated shorter variants while maintaining high Cas9-likelihood and predicted PAMs; independent CICERO evaluation supported PAM preservation.In a fixed-length benchmark, pCoMole achieved 100% PAM match versus 35% for SCISOR and 0% for RayGun; CICERO match rates improved with PAM guidance for St3Cas9 and GeoCas9.Source: §4.3 · pp. 8–9, Table 5 · p. 9

4.4 pCoMole Develops Short Peptidomimetic Binders Optimizing Seven Drug-Related Properties

• Across 15 PDB targets, generated peptidomimetics satisfied terminal constraints and shortened binders while improving predicted permeability, half-life, and affinity, with motif and specificity scores largely preserved or improved.Some targets showed modest decreases in non-toxicity or solubility; docking placed designs in the same binding pockets as the known peptides.Source: §4.4 · p. 9, Table 6 · p. 10
• For clinically relevant peptide drugs, pCoMole produced shorter constraint-satisfying peptidomimetics with improved predicted permeability and half-life while maintaining strong predicted affinity.The case studies covered teriparatide, p28, and semaglutide; semaglutide-derived designs retained favorable predicted profiles relative to reference drugs.Source: §4.4 · pp. 9–10

• Two 229-residue pCoMole eGFP variants retained clear green fluorescence in BL21 cells after 10 deletions.The variants had either one or two substitutions.Source: §5 · p. 10
• Across GFP, Cas9, and peptidomimetic binder tasks, pCoMole shortened sequences while maintaining specified properties and navigating multi-objective trade-offs.Examples include PAM preference and binding motifs.Source: §5 · p. 10
• Rollout-based guidance increases computational cost, while pCoMole provides a principled bridge between discrete generative modeling and biomolecular optimization.Source: §5 · p. 10

A Related Work

• pCoMole addresses variable-length sequence editing with explicit terminal feasibility and controllable multi-objective preferences, unlike cited approaches centered on shrinkage, online evaluation, or other constrained-generation settings.The related-work discussion notes that some alternatives use scalar objectives or soft feasibility handling, while pCoMole samples offline from a fixed pre-trained editor.Source: A Related Work · p. 17

B Log-domain rollout estimation and guidance

• Log-domain computations avoid explicitly exponentiating large utilities and stabilize candidate selection by subtracting the maximum score before exponentiation.The normalized selection is invariant to adding a constant offset to the log-preference or candidate scores.Source: B Log-domain rollout estimation and… · pp. 17–18
• Infeasible terminal states receive zero weight, while the log-mean-exp Monte Carlo value estimate is consistent as the rollout count grows, despite finite-sample bias.When all rollouts are infeasible, the estimated continuation value is negative infinity.Source: B Log-domain rollout estimation and… · pp. 17–18

• With exact harmonic values, Doob-h guidance yields the terminal distribution tilted by the preference function.The result assumes integrability and positive harmonic values on states reachable from the initial state.Source: C Theoretical Proofs · pp. 18–19
• An augmented Tchebycheff utility maximizer is Pareto-optimal among feasible sequences; under stated assumptions, suitable weights recover Pareto-optimal points as Chebyshev-term maximizers.The recovery result additionally assumes objective values exceed their reference values across the feasible set and a positive margin for the augmented term.Source: C Theoretical Proofs · p. 19
• When any feasible terminal is evaluated, pCoMole returns a feasible terminal maximizing preference among evaluated feasible sequences; pruning can reduce empirical coverage under a fixed budget.The incumbent guarantee concerns the evaluated set, not all possible sequences.Source: C Theoretical Proofs · pp. 22–23
• Rollout-based guidance with sufficiently accurate value estimates yields a terminal distribution multiplicatively close to ideal Doob-h guidance over a fixed editing horizon.The result follows from one-step kernel stability and applies to each run with its chosen weight vector.Source: C Theoretical Proofs · p. 23

D Ablation Studies

• Enforcing both peptidomimetic constraints yields short, feasible designs with strong predicted properties and 100% satisfaction; removing both causes length collapse and loss of affinity and motif consistency.Removing LEN alone encourages over-shrinking and worsens trade-offs, while removing PM rarely yields invalid outputs but removes an added safety layer.Source: D Ablation Studies · p. 23
• Augmented Tchebycheff achieves the highest empirical Pareto coverage, versus moderate coverage for standard Tchebycheff and the lowest for linear weighted sums.Changing objective weights produces the expected trade-offs; the comparison evaluates solubility, permeability, and binding affinity for PDB 4O56.Source: D Ablation Studies · p. 24, Table S2 · p. 24
• At 1000× deletion scaling, unguided Edit Flow reaches 0% PAM match, while pCoMole maintains high Cas9-likeness and exact PAM matching at comparable shrinkage.Deletion-rate scaling controls shrinkage pressure, whereas Doob-h guidance controls whether resulting sequences remain feasible.Source: D Ablation Studies · p. 24, Table S3 · p. 24
• Disabling insertions consistently increases Cas9 length reduction with negligible impact on Cas9-likelihood or PAM preservation, making insertion-free editing the preferred shrinkage configuration.The ablation generated 50 shrunk St1Cas9 variants with pCoMole.Source: D Ablation Studies · pp. 24–25, Table S4 · p. 25
• More sampling steps, rollouts, or candidates improve shrinkage quality and predicted properties, especially affinity and motif preservation, but bring diminishing returns and higher runtime.More steps refine progressively, more rollouts reduce myopic decisions at higher cost, and more candidates improve exploration before saturating.Source: D Ablation Studies · p. 25

E.1 Dataset Curation

• UniRefS uses 30k reviewed sequences of length ≤350 with an 80-10-10 train/test/validation split; UniRefL is sampled by Cas9 length buckets, redundancy-reduced, and has about 30k sequences.The UniRefL sampling aims to preserve the Cas9 dataset’s length distribution while making the set as large as possible.Source: §E.1 · p. 25
• The GFP dataset combines FPBase fluorescent proteins in a selected emission range with UniRef PF01353 sequences, removes redundancy, and uses an 80-10-10 split.Source: §E.1 · p. 26
• The Cas9 dataset starts with roughly 105 candidate sequences from the CRISPR-Cas Atlas, then reduces redundancy to around 2 × 10^4 representatives and splits by cluster 80-10-10.Cluster-level splitting keeps highly similar sequences from appearing across the train, validation, and test sets.Source: §E.1 · pp. 26–27
• The peptidomimetic dataset combines a commercial ChemDiv library with peptide-derived candidates generated through peptoid, triazole-isostere, and terminal-deletion edits.Candidates are filtered by synthetic accessibility and SCScore, deduplicated, converted from SMILES to SELFIES, and used for Edit Flow training.Source: §E.1 · p. 27

E.2 Model Architecture

• A Transformer encoder parameterizes a continuous-time Markov chain over discrete edit operations, combining token and flow-time embeddings through Transformer blocks.Rotary positional embeddings support length generalization, and the final normalized representation feeds multiple output heads.Source: §E.2 · p. 27
• Separate heads predict a nonnegative total edit rate and logits over insertion, deletion, and substitution types.The rate uses Softplus; token-level heads produce insertion and substitution vocabulary proposals conditioned on edit type.Source: §E.2 · p. 27
• This factorized parameterization supports per-position edit intensities and token proposals for variable-length sequence generation.Variable length is enabled through insertion and deletion operations.Source: §E.2 · p. 27

E.3 Training Strategy

• Training samples randomized initial sequences, aligns each to a dataset target, and supervises edit dynamics along the continuous-time trajectory.Initial lengths are sampled from task-specific ranges to expose models to varied starting lengths and sequence content.Source: §E.3 · p. 27
• Protein Edit Flow training uses distinct model configurations, while diversity is evaluated on 1,000 generated sequences using alignment-free k-mer Jaccard diversity with k = 3.The UniRefS, UniRefL, and Cas9 models were trained for around 30, 10, and 100 epochs, respectively.Source: §E.3 · pp. 27–28

F.1 Cas9 Score Models

• Cas9 quality scoring uses an MLP classifier over frozen ESM-2-650M embeddings, while PAM matching uses Protein2PAM predictions thresholded at 0.7.The classifier is trained to distinguish Cas9 from length-matched negatives including other Cas effectors and corrupted Cas9-like sequences.Source: §F.1 · p. 28

F.2 GFP Score Models

• GFP family membership is enforced with a classifier trained on the curated GFP dataset, and FPredX predictions define brightness and excitation objectives plus a green-band emission constraint.The classifier’s held-out test AUROC is 0.9784.Source: §F.2 · pp. 28–29

F.3 Peptidomimetics Score Models

• Peptidomimetic developability scores blend PeptiVerse and Admetica predictions according to a candidate’s peptide-likeness, computed from RDKit-derived structural features.The mixture gives greater weight to peptide-specialized predictions for peptide-like candidates and to small-molecule ADMET predictions for less peptide-like candidates.Source: §F.3 · p. 29
• SMILES BindEvaluator fuses SMILES and target-protein representations to predict residue-wise binding sites, using curated binder-target pairs and filtering to limit leakage.The final dataset contains 48,155 training pairs and 6,451 validation pairs.Source: §F.3 · p. 29
• Motif score measures mean predicted binding probability in the known pocket, while specificity counts above-threshold predictions outside the motif and normalizes by sequence length.Higher motif scores indicate stronger predicted binding in the intended region; higher specificity indicates fewer high-confidence predictions outside it.Source: §F.3 · p. 30
• Peptidomimetic feasibility checks classify chemical type and assess shortening through backbone connectivity, molecular weight, heavy-atom count, and a token-length range of [L0/2, L0).Overall shorter-or-smaller status is accepted if the backbone proxy decreases or both molecular weight and heavy-atom count decrease.Source: §F.3 · p. 30

G Independent and Experimental Validation Details

• CICERO evaluated 25 designs per ortholog and condition for St3Cas9 and GeoCas9, with a control removing the PAM objective and hard constraint; Table S8 reports Wilson 95% intervals.The control kept the other pCoMole settings fixed.Source: G Independent and Experimental… · p. 30, Table S8 · p. 31
• Sequence-verified eGFP plasmids were expressed in BL21 cells, retransformed to confirm fluorescence retention, and imaged with 488-nm excitation against original and negative controls.Fluorescence retention was assessed qualitatively.Source: G Independent and Experimental… · p. 31

G.1 Wet-lab eGFP Sequences

• The tested designs included pCoMole_eGFP_1 (229 aa; 10 deletions, 1 substitution) and pCoMole_eGFP_2 (229 aa; 10 deletions, 2 substitutions), alongside a negative control (228 aa; 11 deletions, 2 substitutions).The original eGFP construct is listed as 239 aa.Source: §G.1 · p. 31

H pCoMole Sampling Details.

• GFP sampling used 10 steps, 50 candidates, and 10 rollouts, with insertions disabled and deletion rates amplified 100-fold to encourage shrinkage.The stated settings were intended to encourage aggressive shrinkage while maintaining GFP validity and emission constraints.Source: H pCoMole Sampling Details · pp. 31–32
• Cas9 shrinkage amplified deletion rates and generally disabled insertions; main experiments used 20 steps, 10 candidates, five rollouts, equal objective weights, and 1000-fold deletion scaling.PAM preservation used the input sequence's Protein2PAM-predicted PAM as the target.Source: H pCoMole Sampling Details · p. 32
• Cas9 benchmark variants changed budgets and deletion scaling: compute-matched tests used five steps and 2500-fold scaling, while fixed-length tests used 20 rollouts and extra PAM weight.Insertion and deletion-rate ablations varied those settings as specified in the appendix.Source: H pCoMole Sampling Details · p. 32
• Peptidomimetic design used 30 steps, 30 candidates, 10 rollouts, and seven weighted objectives; sampled SELFIES were converted to SMILES for scoring and constraint checks.Reported average lengths are tokenized SMILES lengths, rounded to integers.Source: H pCoMole Sampling Details · pp. 32–33

I Benchmark Models

• RayGun generations used a fixed public 8.8M-parameter UniRef50 checkpoint, while SCISOR generations used a fixed public UniRef50_L model without further fine-tuning.The appendix reports the GPU systems used for each baseline.Source: I Benchmark Models · p. 33

J Impact Statement

• The impact statement presents smaller Cas9 systems as potentially expanding deliverable genome-editing tools while identifying off-target activity, immunogenicity, and in-vivo behavior as safety concerns.It calls for experimental validation, safety screening, and responsible deployment.Source: J Impact Statement · p. 33
• Table S9 compares pCoMole, SCISOR, and RayGun on WT CjCas9 designs constrained to 934 residues.The caption states the comparison setting but supplies no cell values.Source: Table S9 · p. 34
• Table S10 covers 15 peptide-binder targets, reporting mean scores across 100 designs per target and changes from existing binders; its caption says all designs satisfy specified constraints.Length is measured in tokenized SMILES units, and positive changes indicate improvement.Source: Table S10 · p. 34

K Additional Benchmark Results

• Figure S1 compares average pLDDT distributions for 1,000 UniRefS Edit Flow generations and 1,000 input UniRef sequences of length at most 350.The input sequences were used as generation starting points.Source: Figure S1 · p. 35
• Figure S2 presents GFP variants under four objective-guidance settings, alongside secondary structures, predicted pTM, sequence length, brightness, and excitation-alignment error.The settings vary combinations of brightness, excitation alignment, and length reduction.Source: Figure S2 · p. 35
• Figures S3–S6 show known and designed peptidomimetic binders with target proteins, motif surfaces, interface insets, and predicted property scores across 12 PDB targets.The targets are listed across the figure captions as 1AYC, 1B8Q, 1DDV, 1E6I, 2LTV, 2Q8Y, 4GNE, 5AZ8, 6MLC, 7JVS, 7LUL, and 8CN1.Source: Figure S3 · p. 36, Figure S4 · p. 37, Figure S5 · p. 38
• Figure S7 presents representative shrunk SpCas9, St1Cas9, St3Cas9, and GeoCas9 designs with secondary structures, predicted pTM, length, Cas9-likelihood, and PAM.The caption identifies these four orthologs but does not report comparative outcomes.Source: Figure S7 · p. 40

Claims and scope

• The abstract and introduction accurately present pCoMole’s contributions: constrained Pareto-aligned editing, Doob-h guidance, Monte Carlo rollouts with guarantees, and evaluations across three biomolecular tasks.The cited tasks are GFP and Cas9 shrinkage and peptidomimetic binder design.Source: NeurIPS Paper Checklist · p. 41

Limitations

• pCoMole’s higher computational cost than shrinkage-only baselines arises from scoring candidate edits and estimating continuation values with Monte Carlo rollouts; larger hyperparameter budgets show diminishing returns.Runtime scales with the number of steps, candidates per step, and rollouts per candidate; results are validated in silico, with wet-lab validation identified as a next step.Source: NeurIPS Paper Checklist · p. 41

Theory assumptions and proofs

• All theoretical results are formally stated and proved in Appendix C, covering the Doob-h transform, Pareto coverage, rollout approximation, error conversion, multistep stability, and incumbent feasibility.Appendix B provides log-domain rollout estimation details.Source: NeurIPS Paper Checklist · p. 42

Experimental result reproducibility

• The paper reports full reproduction details for training and sampling across GFP, Cas9, and peptidomimetic tasks, and provides an anonymous code repository.Appendices describe training procedures, datasets, architectures, optimizer settings, and task-specific sampling hyperparameters.Source: NeurIPS Paper Checklist · p. 42

Societal impact and safeguards

• The Impact Statement discusses potential benefits of smaller genome-editing systems and risks including off-target activity, immunogenicity, and context-dependent behavior, calling for validation and safety screening.The paper reports that it releases no new high-risk foundation models, datasets, or generators.Source: NeurIPS Paper Checklist · pp. 45–46

NeurIPS Paper Checklist

• LLM usage is marked N/A; the method uses a pre-trained ESM-2-650M sequence encoder for protein embeddings, not LLMs as a core component.The framework uses a Transformer-based Edit Flow backbone, but the checklist does not classify it as an LLM component.Source: NeurIPS Paper Checklist · p. 48